Solve these systems of equations using the Substitution Method
x = 3y − 5
2x + 12y = −4
x = –4, y = 1/3
y = 3x + 7
y = −4x + 21
x = 2, y = 13
3x − y = 17
−x + y = −7
x = 5, y = –2
A method for solving a system of equations by replacing one variable with an expression involving the remaining variable(s). For example, in the system of equations below, the first equation tells you that y is equal to –3x + 5. We can _____ –3x + 5 in for y in the second equation to get 2(–3x + 5) + 10x = 18, then solve this equation to find x. Once we have x, we _____ that value back into either of the original equations to find the value of y.
y = –3x + 5
2y + 10x = 18
2x − y = 10
y = −4x + 2
y = −x + 8
y = x − 2
2x + 3y = 9
−3x + 3y = −6
A method for solving a system of equations. The key step in using the _____ is to add both sides of two equations to _____ one of the variables. For example, the two equations in the system below can be added together to get the simplified result 7x = 14. We can solve this equation to find x, then substitute the x-value back into either of the original equations to find the value of y.
5x + 2y = 10
2x – 2y = 4
x = 8 − 2y
y − x = 4
y = −1/2x + 7
y = x − 8
9x + 10y = 14
7x + 5y = −3
A method for solving a system of equations. To use the _____, take two expressions that are each equal to the same variable and set those expressions equal to each other. For example, in the system of equations below, –2x + 5 and x – 1 each equal y. So we write –2x + 5 = x – 1, and then solve that equation to solve for x. Once we have x, we substitute that value back into either of the original equations to find the value of y.
y = –2x + 5
y = x – 1
x = −2y − 3
4y − x = 9
y = 1/4x + 5
y = 2x − 9
x + 5y = 8
−x + 2y = −1
A _____ is written at the beginning of our work to identify the variable that will represent a certain quantity. For example, in solving a problem about grilled cheese sandwiches, we might begin by writing “_____ s = the number of sandwiches eaten.” It is particularly important to use _____ when writing mathematical sentences, so that your readers will know what the variables in the sentences represent.
y = 1/3x + 4
x = −3y
−2x + 3y = 1
2x + 6y = 2
A _____ is a set of equations with the same variables. Solving a _____ means finding one or more solutions that make each of the equations in the _____ true. A solution to a _____ gives a point of intersection of the graphs of the equations in the _____. There may be zero, one, or several solutions to a _____.